Parameter Identification in One-Dimensional Partial Differential Algebraic Equations

K. Schittkowski, GAMM-Mitteilungen, Vol. 30, No. 2, 352-375  (2007)
Abstract: In this paper we discuss a couple of situations, where algebraic equations are to be attached to a system of one-dimensional partial differential equations.
Besides of models leading directly to algebraic equations because of the underlying practical background, for example in case of steady-state equations, there are many others where the specific mathematical structure requires a certain reformulation leading to time-independent equations. To be able to apply our approach to a large class of real-life problems, we have to take into account flux formulations, constraints, switching points, different integration areas with transition conditions, and coupled ordinary differential algebraic equations (DAEs), for example. The system of partial differential algebraic equations (PDAEs) is discretized by the method of lines leading to a large system of differential algebraic equations which can be solved by any available implicit integration method. Standard difference formulas are applied to discretize first and second partial derivatives, and upwind formulae are used for transport equations. Proceeding from given experimental data, i.e., observation times and measurements, the minimum least squares distance of measured data from a fitting criterion is computed, which depends on the solution of the system of PDAEs. Parameters to be identified can be part of the differential equations, initial, transition, or boundary conditions, coupled DAEs, constraints, fitting criterion, etc. Also the switching points can become optimization variables. The resulting least squares problem is solved by an adapted sequential quadratic programming (SQP) algorithm which retains typical features of a classical Gauss-Newton method by retaining robustness and fast convergence speed of SQP methods. The mathematical structure of the identification problems is outlined in detail, and we present a number of case studies to illustrate the different model classes which can be treated by our approach.

To download a preprint, click here: pdaerev.pdf


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